Asymptotics of singular values for quantum derivatives

Abstract

We obtain Weyl type asymptotics for the quantised derivative $\dbar f$ of a function $f$ from the homgeneous Sobolev space ${\dot{W}}_d^1({\mathbb{R}}^d)$ on ${\mathbb{R}}^d.$ The asymptotic coefficient $\Vert \nabla f{\Vert }_{L_d({\mathbb{R}}^d)}$ is equivalent to the norm of $\dbar f$ in the principal ideal ${\mathcal{L}}_{d,\infty },$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dbar f.$ Our methods are based on the $C^{\ast }$-algebraic notion of the principal symbol mapping on ${\mathbb{R}}^d$, as developed recently by the last two authors and collaborators.

41 pages; accepted for publication in Trans. Amer. Math. Soc